It’s interesting how a conversation with Bessis can lead to so many different views.
I think concepts are important, but I don’t think mathematics is fundamentally about concepts.
Concepts are how we enter mathematics.
Constraints are what mathematics is about.
A concept can be invented.
A constraint cannot.
Nobody invented rotational symmetry, conservation laws, prime structure, or geometric invariants. We discovered them.
The reason mathematics is so effective in physics has nothing to do with the universe resembling our concepts. It’s because both mathematics and physics are ultimately constrained by the same underlying structures.
For me, mathematics is not the study of concepts. It is the study of transformation under necessary constraints.
Concepts come and go.
What survives constraint is what mathematics is actually revealing.
I agree substantially with what you are saying, especially that we discovered all of these mathematical structures. My question though is where the constraints come from.
My view is that the constraints are inherent in the foundational concepts once we follow them through rigorously. Start with the concept of the number 1 and addition, and everything we know about the natural numbers, and prime numbers, starts to unfold.
Mathematics isn't about so much trying to understand concepts, but it is more identifying what follows necessarily from certain concepts under the constraints both inherent in the concept and the constraints of logic.
It’s interesting how a conversation with Bessis can lead to so many different views.
I think concepts are important, but I don’t think mathematics is fundamentally about concepts.
Concepts are how we enter mathematics.
Constraints are what mathematics is about.
A concept can be invented.
A constraint cannot.
Nobody invented rotational symmetry, conservation laws, prime structure, or geometric invariants. We discovered them.
The reason mathematics is so effective in physics has nothing to do with the universe resembling our concepts. It’s because both mathematics and physics are ultimately constrained by the same underlying structures.
For me, mathematics is not the study of concepts. It is the study of transformation under necessary constraints.
Concepts come and go.
What survives constraint is what mathematics is actually revealing.
I agree substantially with what you are saying, especially that we discovered all of these mathematical structures. My question though is where the constraints come from.
My view is that the constraints are inherent in the foundational concepts once we follow them through rigorously. Start with the concept of the number 1 and addition, and everything we know about the natural numbers, and prime numbers, starts to unfold.
Mathematics isn't about so much trying to understand concepts, but it is more identifying what follows necessarily from certain concepts under the constraints both inherent in the concept and the constraints of logic.
I won’t do the math in here but no. I don’t think it works that way. Interesting thought though.
Thanks. I'd be interested to see the math, but obviously this isn't the right place for it.
But I'm more interested in where you think the constraints you mentioned come from. I'm trying to understand how your view fits together.